Lesson 6 of 6 · 12 min

FX forwards: carry as an interest rate differential

For currencies, the cost of carry is the gap between the two interest rates, so the FX forward rate is the spot rate grown at the price currency's rate minus the base currency's rate.

In short

  • An FX quote Sf/dS_{f/d} is the number of units of the price currency (f) per one unit of the base currency (d). USD/EUR = 1.10 means USD 1.10 per EUR 1.
  • A long FX forward buys the base currency and sells the price currency at a rate fixed today.
  • No-arbitrage: F0,f/d(T)=S0,f/d e(rf−rd)TF_{0,f/d}(T) = S_{0,f/d}\, e^{(r_f - r_d)T}. Only the rate differential matters, not the level of rates.
  • rf>rdr_f > r_d: F > S, the price currency trades at a forward discount. rf<rdr_f < r_d: F < S, the price currency trades at a forward premium.
  • Replication: borrow the base currency, convert at spot, lend the price currency; the ratio of the two balances at T is the forward rate.

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FX forwards: carry as an interest rate differential · Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives